Generating solutions of "impossible-to- solve" problems and simulating "impossible-to-simulate" models Florent Krzakala Espci - Paristech L. Zdeborová ( Los Alamos, LANL ) http://www.pct.espci.fr/~florent fl @espci.fr
Planting ! Florent Krzakala Espci - Paristech L. Zdeborová ( Los Alamos, LANL ) http://www.pct.espci.fr/~florent fl @espci.fr
Impossible-to-solve problems Some optimization problems such as COL and SAT are almost impossible to solve! ex: Hard Instances of random graph coloring Text A 3 - coloring of a random graph with c=4.36
Impossible-to-solve problems Some optimization problems such as COL and SAT are almost impossible to solve! ex: Hard Instances of random graph coloring The “ ” problem q log q Text A 3 - coloring of a random graph with c=4.36 D. Achlioptas et al. Nature 2005
Impossible-to-solve problems Some optimization problems such as COL and SAT are almost impossible to solve! ex: Hard Instances of random graph coloring The “ ” problem q log q • Consider q color ( with q large enough ) and a large random graph of average degree c • W .h.p this graph is colorable if c<2q log q Text • However, no algorithm is able to do so e ffi ciently ( polynomial ) for c> q log q ! A 3 - coloring of a random graph with c=4.36 D. Achlioptas et al. Nature 2005 Averag e q log q 2q log q degree c
Impossible-to-solve problems Some optimization problems such as COL and SAT are almost impossible to solve! ex: Hard Instances of random graph coloring The “ ” problem q log q • Consider q color ( with q large enough ) and a large random graph of average degree c • W .h.p this graph is colorable if c<2q log q Text • However, no algorithm is able to do so e ffi ciently ( polynomial ) for c> q log q ! A 3 - coloring of a random graph with c=4.36 D. Achlioptas et al. Nature 2005 Averag e q log q 2q log q degree c COL UNCOL
Impossible-to-solve problems Some optimization problems such as COL and SAT are almost impossible to solve! ex: Hard Instances of random graph coloring The “ ” problem q log q • Consider q color ( with q large enough ) and a large random graph of average degree c • W .h.p this graph is colorable if c<2q log q Text • However, no algorithm is able to do so e ffi ciently ( polynomial ) for c> q log q ! A 3 - coloring of a random graph with c=4.36 D. Achlioptas et al. Nature 2005 Averag e q log q 2q log q degree c COL UNCOL Possibl e Impossibl e
Impossible-to-solve problems Some optimization problems such as COL and SAT are almost impossible to solve! ex: Hard Instances of random graph coloring The “ ” problem q log q • Consider q color ( with q large enough ) and a large random graph of average degree c • W .h.p this graph is colorable if c<2q log q Text • However, no algorithm is able to do so e ffi ciently ( polynomial ) for c> q log q ! A 3 - coloring of a random graph with c=4.36 No one has ever seen the solution of, say 5 - coloring, for large enough c and N=10 6 D. Achlioptas et al. Nature 2005 Averag e q log q 2q log q degree c COL UNCOL Possibl e Impossibl e
Impossible-to-simulate problems Some optimization problems such as COL and SAT are also hard to sample Averag e q log q 2q log q degree c
Impossible-to-simulate problems Some optimization problems such as COL and SAT are also hard to sample � H = δ ( s i , s j ) Consider the fo � owing “coloring” or “Potts - Antiferromagnet” <ij> Hamiltonia n s i = 1 , 2 , . . . , q Averag e q log q 2q log q degree c
Impossible-to-simulate problems Some optimization problems such as COL and SAT are also hard to sample � H = δ ( s i , s j ) Consider the fo � owing “coloring” or “Potts - Antiferromagnet” <ij> Hamiltonia n Temperatur e s i = 1 , 2 , . . . , q Dynamic transitio n Averag e q log q 2q log q degree c
Impossible-to-simulate problems Some optimization problems such as COL and SAT are also hard to sample � H = δ ( s i , s j ) Consider the fo � owing “coloring” or “Potts - Antiferromagnet” <ij> Hamiltonia n Temperatur e s i = 1 , 2 , . . . , q Dynamic transitio n Possible - to - sample regio n Impossible - to - sample regio n Averag e q log q 2q log q degree c
Frustrating Intractable Problems W e know that some random problems DO have solutions, but we cannot find them! Sampling and performing Monte - Carlo is even Harder! Many predictions from statistical physics in random problems.... but impossible to test most of them !
A Different “Reverse” strategy Problem Solution
A Different “Reverse” strategy Problem Solution Solution Problem
A Different “Reverse” strategy Problem Solution Solution Problem Instead of choosing a problem, and looking for a solution....
A Different “Reverse” strategy Problem Solution Solution Problem Instead of choosing a problem, and looking for a solution.... We choose a configuration/assignment and and look for a problem for which this is a solution !
The Planted Ensemble in the coloring problem Consider the 3 - coloring problem with N nodes and M links. 1 ) Color randomly the N nodes
The Planted Ensemble in the coloring problem Consider the 3 - coloring problem with N nodes and M links. 1 ) Color randomly the N nodes 11 ) Put the M links randomly such that the planted configuration is a proper coloring
The Planted Ensemble in the coloring problem Consider the 3 - coloring problem with N nodes and M links. 1 ) Color randomly the N nodes 11 ) Put the M links randomly such that the planted configuration is a proper coloring 111 ) Now, we have created a problem for which we know the solution
The Planted Ensemble in the coloring problem Consider the 3 - coloring problem with N nodes and M links. 1 ) Color randomly the N nodes 11 ) Put the M links randomly such that the planted configuration is a proper coloring 111 ) Now, we have created a problem for which we know the solution IV ) W e could also have prepared a configuration with a known cost/energy
The Random ensemble versus the Planted ensemble Random ensembl e Planted ensembl e Choose a random graph Choose a random coloring of N with N nodes and M links nodes Choose a random graph such that this is a correct coloring...
The Random ensemble versus the Planted ensemble Random ensembl e Planted ensembl e Choose a random graph Choose a random coloring of N with N nodes and M links nodes Choose a random graph such that this is a correct coloring... Is it really the same to look for a solution a random problem and to look for a random problem that matches a random solution ?
The Random ensemble versus the Planted ensemble Random ensembl e Planted ensembl e Choose a random graph Choose a random coloring of N with N nodes and M links nodes Choose a random graph such that this is a correct coloring... Is it really the same to look for a solution a random problem and to look for a random problem that matches a random solution ? The surprising answer is: in some cases YES !
The Random ensemble versus the Planted ensemble Random ensembl e Planted ensembl e Choose a random graph Choose a random coloring of N with N nodes and M links nodes Choose a random graph such that this is a correct coloring... Is it really the same to look for a solution a random problem and to look for a random problem that matches a random solution ? The surprising answer is: in some cases YES ! Montanari and Semerjian, Jstat. ‘06 & Achlioptas and Coja-Oghlan, arXiv:0803.2122: The two ensembles are asymptotically (N ➔ ∞ ) equivalent for low enough degree c !
The Random ensemble versus the Planted ensemble Random ensembl e Planted ensembl e Choose a random graph Choose a random coloring of N with N nodes and M links nodes Choose a random graph such that this is a correct coloring... Definition : Two ensembles of random graphs are asymptotically equivalent if and only if in the thermodynamic limit every property which is almost surely true on a graph from one ensemble is also almost surely true on a graph from the other ensemble.
Some open questions: Until which connectivity/degree c the planted and random ensembles are equivalent ? Is the planted ensemble interesting beyond this connectivity ? Can we generalize this approach to finite energy ( coloring with a finite fraction of mistakes ? ) How can we use a planted configuration ? What are the models where a “quiet” planting is possible ?
In this talk: 1 ) A ( brief ) summary of a theory of “quiet” planting in random models 2 ) Using planted configurations for fast simulations.
The Planted Ensemble
The Planted Ensemble * W e use the formalism described in Zdeborová’s talk
Main result Consider a model where the annealed computation is correct in some region ( high temperature or low degree ) f = − 1 f annealed = − 1 N β [log Z ] dis N β log [ Z ] dis
Main result Consider a model where the annealed computation is correct in some region ( high temperature or low degree ) f = − 1 f annealed = − 1 N β [log Z ] dis N β log [ Z ] dis = Consider a model where a factorized ( i.e. identical for all nodes ) Belief Propagation solution is correct in some region ( high temperature or low degree )
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